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\(\partial\)-reducing Dehn surgeries and 1-bridge knots. (English) Zbl 0788.57005
Suppose \(K\) is a knot in a \(\partial\)-reducible 3-manifold \(M\) such that no compressing disk of \(\partial M\) is disjoint from \(K\). A Dehn surgery on \(K\) is called \(\partial\)-reducing if the surgered manifold is \(\partial\)-reducible, \(K\) is called \(\partial\)-reducing if it admits such a surgery. The 1-bridge conjecture states that if a knot \(K\) is a \(\partial\)-reducing knot, then either \(K\) or its dual knot is a 1-bridge knot.
The results of this paper are two-sided. It is first shown that the conjecture is true if there is an essential torus separating the knot from \(\partial M\). As a consequence, the author gives a complete classification of all satellite \(\partial\)-reducing knots. In the second part the author gives a counterexample to the 1-bridge conjecture and its weakened version, thus settling the problem in general. The third part is a study of the still open problem of whether the conjecture is true for knots in handlebodies. It is shown that if \(K\) is a 1-tunnel \(\partial\)- reducing knot in a handlebody, then it is a 1-bridge knot. It is also shown that if \(K\) is a 1-tunnel \(\partial\)-reducing knot, then its dual knot is also a 1-tunnel knot. A corollary of this result is: If a Dehn surgery on a 1-bridge knot in a handlebody yields a handlebody, then the dual knot corresponding to this surgery is also a 1-bridge knot.
Reviewer: Y.-Q.Wu (Austin)

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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